I have made some customs PCBs with ISP headers for programming with AS7. With the USBtiny ISP(ebay clone) I have successfully programmed the atmega328's but NOT the atmega2561. I found out that this happens because of its large flash memory(256KMB). If this is correct, could you propose me other cheap solution? (below 50$)

The size of the flash should make no difference. Are you using a very old version of software to drive the USBtiny ISP on the PC? If you simply get a $2 USBAsp from ebay and drive it with avrdude there should be no issue with programming 256K devices.

I tried to program an arduino MEGA(atmega2560) and works fine. As concerning the drivers I reinstall the newest version I found. But I still can't program the atmega2561. I checked the connections, no shortcuts, everything is ok.

Unless you used a blowtorch, it is unlikely that you damaged the chip. I’d suggest you check the wiring and ensure you’ve used the correct pins for programming - check the datasheet. The usual error is to use the mosi and miso pins like you would with a mega328.

You say you have problems with the atmega2561. Does this chip use the standard SPI interface for programming or does it use one of the other "more modern" interfaces?

I have only used AVR's which are compatible with the USBap software myself.

Intel Hex files are limited to 64kB address space. If you go over that, then there is a workaround in the Intel Hex format for "address records" to do something similar to bank switching. It could be that this leads to problems if a part of your software chain does not support this properly.

A possible workaround is to use the Motorola Srecord file format or binary format to send files to your programmer.

The largest unknown seems to be your USBtinyisp programmer. You could try some other firmware for it or use an USBasp (based on atmega8).

And it literary says that it is not compatible with AVR's with > 64kB Flash.

What chips can be programmed?
Any AVR that uses the ISP interface for programming and has 64K or less of flash can be programmed.
Chips such as the Atmega1280/1281 and Atmega2560/2561 have more than 64K and cannot be programmed.